Efficient Splitting and Merging Algorithms for Order Decomposable Problems

Let S be a set whose items are sorted with respect to d > 1 total orders 1 ; : : : ; d , and which is subject to dynamic operations, such as insertions of a single item, deletions of a single item, split and concatenate operations performed according to any chosen order i (1 i d). This generalizes to dimension d > 1 the notion of concatenable data structures, such as the 2-3-trees, which support splits and concatenates under a single total order. The main contribution of this paper is a general and novel technique for solving order decomposable problems on S, which yields new and eecient concatenable data structures for dimension d > 1. By using our technique we maintain S with the following time bounds: O(log n) for the insertion or the deletion of a single item, where n is the number of items currently in S; O(n 1?1=d) for splits and concatenates along any order, and for rectangular range queries. The space required is O(n). We provide several applications of our technique. Namely, we present new multidi-mensional data structures implementing two-dimensional priority queues, two-dimensional search trees, and concatenable interval trees: these data structures allow us to improve many previously known results on decomposable problems under split and concatenate operations, such as membership query, minimum-weight item, range query, convex hulls and Voronoi diagrams.